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Maximal regularity estimates for the abstract Cauchy problems

Sebastian Król, Mieczysław Mastyło, Jarosław Sarnowski

Abstract

In this work, we extend the Da Prato-Grisvard theory of maximal regularity estimates for sectorial operators in interpolation spaces. Specifically, for any generator $-A$ of an analytic semigroup on a Banach space $X$, we identify the interpolation spaces between $X$ and the domain $D_{A}$ of $A$ in which the part of $A$ satisfies certain maximal regularity estimates. We also establish several new results concerning both homogeneous and inhomogeneous $L^1$-maximal regularity estimates, extending and completing recent findings in the literature. These results are motivated not only by applications to problems in areas such as fluid mechanics but also by the intrinsic theoretical interest of the subject. In particular, we address the optimal choice of data spaces for the Cauchy problem associated with $A$, ensuring the existence of strong solutions with global-in-time control of their derivatives. This control is measured via the homogeneous parts of the interpolation norms in the spatial variable and weighted Lebesgue norms over the time interval. Furthermore, we characterize weighted $L^1$-estimates and establish their relationship with unweighted estimates. Additionally, we reformulate the characterization condition for $L^1$-maximal regularity due to Kalton and Portal in a priori terms that do not rely on semigroup operators. Finally, we introduce a new interpolation framework for $L^p$-maximal regularity estimates, where $p \in (1, \infty)$, within interpolation spaces generated by non-classical interpolation functors.

Maximal regularity estimates for the abstract Cauchy problems

Abstract

In this work, we extend the Da Prato-Grisvard theory of maximal regularity estimates for sectorial operators in interpolation spaces. Specifically, for any generator of an analytic semigroup on a Banach space , we identify the interpolation spaces between and the domain of in which the part of satisfies certain maximal regularity estimates. We also establish several new results concerning both homogeneous and inhomogeneous -maximal regularity estimates, extending and completing recent findings in the literature. These results are motivated not only by applications to problems in areas such as fluid mechanics but also by the intrinsic theoretical interest of the subject. In particular, we address the optimal choice of data spaces for the Cauchy problem associated with , ensuring the existence of strong solutions with global-in-time control of their derivatives. This control is measured via the homogeneous parts of the interpolation norms in the spatial variable and weighted Lebesgue norms over the time interval. Furthermore, we characterize weighted -estimates and establish their relationship with unweighted estimates. Additionally, we reformulate the characterization condition for -maximal regularity due to Kalton and Portal in a priori terms that do not rely on semigroup operators. Finally, we introduce a new interpolation framework for -maximal regularity estimates, where , within interpolation spaces generated by non-classical interpolation functors.

Paper Structure

This paper contains 16 sections, 41 theorems, 262 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The maximal regularity
  4. The mild solutions
  5. The characterization of weighted $L^1$-m.r.
  6. The K-method
  7. The Hardy operator
  8. The representation of interpolation spaces $K_\Phi({\mathcal{X}},D_{\mathcal{A}})$
  9. Equivalent representation of homogenous part of interpolation norms
  10. The density of the domain in interpolation spaces
  11. Extension of Da Prato - Grisvard theorem
  12. An extension of the Da Prato-Grisvard theorem for $L^1$-m.r.
  13. An extension of the Da Prato-Grisvard theorem for $L^\infty$-m.r.
  14. Maximal regularity estimates via completion and extrapolation
  15. The interpolation approach
  16. ...and 1 more sections

Key Result

Lemma 2.2

Let $-A$ be the generator of a semigroup $T$ on a Banach space $X$. Let $f\in L^1_{loc}([0,\infty);X)$ and set The following assertions hold.

Theorems & Definitions (89)

  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3: Ha06
  • proof : Proof of Lemma $\ref{['regularity of u']}$
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • ...and 79 more